A High-Order Finite-Volume Scheme for the Dynamical Core of Weather and Climate Models

ChristianeJablonowski and Paul A. Ullrich, University of Michigan, Ann Arbor

Overview

Tests of the Nonhydrostatic Model at all Scales

Cubed-Sphere Grids

Idealized tests in Cartesian geometry that assess the model performance at the microscale(2D bubble resembling convection), mesoscale (2D mountain waves) and 3D global scale:

The future generation of atmospheric models used for weather and climate predictions will likely rely on both high-order accuracy and Adaptive Mesh Refinement(AMR) techniques in order to properly capture the atmospheric features of interest. We present our ongoing research on developing a set of conservative and highly accurate numerical methods for simulating the atmospheric fluid flow (the so-called dynamical core).

In particular, we have developed a fourth-order finite-volume scheme for a nonhydrostatic dynamical core on a cubed-sphere grid (Ullrich and Jablonowki 2010, 2011a,b)that makes use of an implicit-explicit Runge-Kutta-Rosenbrock (IMEX-RKR) time integrator and Riemann solvers. We survey the algorithmic steps, present results from idealized dynamical core test cases and outline the inclusion of AMR into the model design.

The cubed-spheregrid consists of 6 faces (in red) that are subdivided (in blue). The grid point distribution is almost uniform. We use a non-ortho-gonal equiangularcubed-sphere grid.

We collaborate with Phillip Colella (Lawrence Berkeley National Laboratory) and will soon introduce an Adaptive Mesh Refinement (AMR) Technique via the Chombo library.

Algorithmic Design

High-Order Sub-Grid Reconstructions and Source Terms

High-order accuracy can be achieved in finite-volume methods using high-degree polynomials to locally reconstruct the prognostic variables in each cell. We make use of cubic polynomials to ensure our schemes achieve fourth-order accuracy. We utilize the convolution / deconvolutionmethod by Barad and Colella (2005) to compute source terms and fluxes with fourth-order accuracy.

Tests of the Nonhydrostatic Model (MCORE) onthe Cubed-Sphere

Intercomparison to the dynamical cores FV, FVcubed, Eulerian and HOMME that are part of NCAR’s Community Atmosphere Model

MCORE

We test our nonhydrostatic dynamical core on the cubed-sphere grid (MCORE, left) with the baroclinic wave test by Jablonowski and Williamson (2006). The right figure shows intercomparison plots from four NCAR dynamical cores. The high-order accuracy of MCORE suppresses the grid imprinting seen in 2nd-order cubed-sphere models like FVcubed. The finite-volume scheme also prevents artifacts like spectral ringing (noise) seen in EUL and HOMME.

Low-Speed Riemann Solver

Numerical fluxes obtained from a reconstruction-based approach are computed using a Riemann solver. We have adapted a version of the low-diffusion AUSM+-up Riemann solver (Liou 2006), which was designed for aerospace applications at low Mach numbers.

Implicit-Explicit Runge-Kutta-Rosenbrock Time Integrator

3D atmospheric models have a large horizontal grid spacing (10-400 km) compared to a relatively short vertical grid spacing (10-1000m). Explicit numerical integration relies on time steps which are proportional to the grid size, so the maximum time step is determined by the highly restrictive vertical grid spacing. Our scheme couples a2nd-order implicit method in the vertical with a 4th-order explicit RK method in the horizontal to relax the CFL restriction in the vertical, lengthen the time stepand improve the computational performance.

Model MCORE: Surface pressure in hPa at day 9. The grid spacing isapprox. 1x1 deg, with 26 levels.

Grid spacing: 1x1 deg, with 26 levels

References

Future Aqua-PlanetTests with Physics Parameterizations

Barad, M. and P. Colella, 2005: A fourth-order accurate local refinement method for Poisson’s equation. J. Comput. Phys., 209, 1-18

Colella, P., D. T. Graves, N. D. Keen, T. J. Ligocki, D. F. Martin, P. W. McCorquodale, D. Modiano, P. O.Schwartz, T. D. Sternberg, and B. V. Straalen, 2009: Chombo Software Package for AMR ApplicationsDesign Document

Jablonowski and Williamson, 2006: A baroclinicinstabilitiy test case for atmospheric model dynamical cores. Quart. J. Roy. Meteor. Soc., 132 (621C), 2943-2975

Liou, M.-S., 2006: A sequel to AUSM, Part II: AUSM+-up for all speeds. J. Comput. Phys., 214, 137-170

Reed, K. A. and C. Jablonowski, 2011: An analytic vortex initialization technique for idealized tropical cyclone studies in AGCMs, Mon. Wea. Rev.,139, 689-710

Ullrich, P. and C. Jablonowski, 2011a: Operator-split Runge-Kutta-Rosenbrock (RKR)methods for non-hydrostatic atmospheric models. Mon. Wea. Rev., in revision

Ullrich, P. and C. Jablonowski, 2011b: MCore: A non-hydrostatic atmospheric dynamical core utilizing high-order finite-volume methods, J. Comput. Phys., in prep. for submission in May 2011

Ullrich, P. A., C. J. Jablonowski, and B. L. van Leer, 2010: High-order finite-volume models for the shallow-water equations on the sphere. J. Comput. Phys., 229, 6104-6134

We have developed a tropical cyclone test case (Reed and Jablonowski, 2011) and will test MCORE with moist physics parameterizations.

Idealized tropical cyclone simulations with NCAR’s dynamical cores. The wind speeds at day 10 differ significantly.